Table of contents

The concept of a Prüfer ring is generalized to orders in simple Artinian rings so that the new concept gives a minimal class of rings closed under Morita equivalence, but in the commutative case does not extend the class of Prüfer domains. In §1 this problem is solved and some elementary properties of noncommutative Prüfer rings are given. In §2 theorems on the localization of a noncommutative Prüfer ring with respect to a prime ideal are proved, these being the basis of the theory. In §3 noncommutative Prüfer rings in a simple finite-dimensional algebra over a field are considered. The main problem, which is posed and partially solved here, involves the connection between a noncommutative Prüfer ring and its center.

An investigation is made of the geometry of the multiplication mappings for monads whose functorial parts are (weakly) normal (in the sense of Shchepin) functors acting in the category of compacta. A characterization is obtained for a power monad as the only normal monad such that the multiplication mapping is soft for some . It is proved that the multiplication mappings and of the inclusion hyperspace monad and the monad of complete chained systems are homeomorphic to trivial Tychonoff fibrations for openly generated continua that are homogeneous with respect to character.

An infinitesimal description is obtained for fractional iterates of analytic functions on the unit disk under the condition that the functions and their iterates do not move fixed points on the unit circle at which they have finite angular derivatives.

It is determined under what conditions the standard problem of extension of a mapping to the whole space is solvable for any closed subset . For finite-dimensional metric compacta and -complexes this is equivalent to the system of inequalities -. The result is applied to finding conditions for general position of a compactum in a Euclidean space.

A refinement is given for a result of M. S. Bazelkov on the exact constant of interpolation of the class of continuous functions with a given convex majorant of the modulus of continuity by Bernstein polynomials.

Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.

A semigroup is called filtering if each of its subsemigroups has the smallest (with respect to inclusion) generating set. It is proved in this article that every maximal chain of nonempty subsemigroups of a finite filtering semigroup has length equal to the order of the semigroup, and that filtering semigroups are characterized by this property in the class of finite semigroups. The main result is a characterization of the class of finite filtering semigroups by means of forbidden divisors, to which end the author finds all finite nonfiltering semigroups all of whose proper divisors are filtering semigroups.

It is proved that any prime Malcev superalgebra of characteristic 2, 3 with nonzero odd part is a Lie superalgebra.

The set of Lebesgue points of a locally integrable function on -dimensional Euclidean space , , is an -set of full measure. In this article it is shown that every -set of full measure is the set of Lebesgue points of some measurable bounded function, and, further, that a set with these properties is the set of points of convergence and nontangential (stable) convergence of a singular integral of convolution type:

for some measurable bounded function . On the basis of this result the set of points of summability of a multiple Fourier series by methods of Abel, Riesz, and Picard types is described.

Let be continuous mappings of a compactum onto compacta , . The following theorem is known for : if any bounded function on can be represented in the form , where and are bounded functions on and , then any continuous can be represented in the same form with continuous and . An example is constructed showing that the analogous theorem is false for .

Conditions are presented under which the relative index of a critical set realizing a local minimum of a nonsmooth functional coincides with the Euler-Poincaré characteristic of this set. An analogous result is obtained for the index of a functional increasing at .

A special commutative Moufang loop of order is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type . Next, with the help of , a 27-dimensional representation is constructed for over , . This makes it possible to prove anew the embedding . A similar construction concerning the embedding is described.

Let be a finite primitive linear group. We prove that if contains a normal subgroup of order 32 then the quotient variety is birationally isomorphic to , where is the Segre cubic. We also prove the rationality of for a large class of such groups (in particular, solvable groups).

Best-possible estimates are obtained for solutions of the Dirichlet problem for linear elliptic nondivergence equations of second order near a conical point of the boundary of the domain under minimal conditions on the smoothness of the coefficients of the equation.

This article is devoted to series in the Walsh-Paley system. In particular, for an integrable function a condition is obtained for p-strong summability of the Fourier-Walsh series a.e. on [0,1] along with a uniform analogue for a continuous, function on [0,1].

Well-posed solvability is proved in an appropriate energy space of a boundary value problem with a nonlocal boundary condition for a one-dimensional parabolic equation; two-sided uniform estimates of the solution are obtained, which replace the maximum principle. The existence of an optimal control of the diffusion coefficient in the problem of minimizing the quality functional is established in the class of functions of bounded variation.

Problems of approximation in a class of function spaces, including Sobolev spaces, by subspaces of finite-element type generated by translations of a lattice of given functions are considered. Widths that describe the approximation properties of such subspaces are defined, and their exact values are enumerated. Necessary and sufficient conditions are obtained for the optimality of subspaces on which these widths are realized. Criteria for the optimality of lattices in terms of the density of lattice packings of certain functions (for Sobolev spaces, of densities of packings by identical spheres) are established. Problems of comparison of the widths used in this article with the Kolmogorov widths of the same mean dimension are discussed.